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MATH1111 Quizzes

Anti-differentiation (analytically) Quiz
Web resources available Questions

This quiz tests the work covered in the lecture on the analytical interpretation of the anti-derivative and corresponds to Section 6.2 of the textbook Calculus: Single and Multivariable (Hughes-Hallett, Gleason, McCallum et al.).

There are more web quizzes at Wiley, select Section 2.

There are quizzes and web links on this topic at http://quiz.econ.usyd.edu.au/mathquiz/integration/index.php

Which of the following statements are correct? (Zero or more options can be correct)
a)
x2dx = 2x + c
b)
2x3dx = 1 2x4 + c
c)
(x2 + 3x + 1)dx = 1 3x3 + 3x2 + x + c
d)
sinxdx = cosx + c
e)
cosxdx = sinx + c

There is at least one mistake.
For example, choice (a) should be False.
2x is the derivative of x2.
There is at least one mistake.
For example, choice (b) should be True.
d dx 1 2x4 + c = 4 2x3 = 2x3
There is at least one mistake.
For example, choice (c) should be False.
d dx 1 3x3 + 3x2 + x + c = x2 + 6x + 1 so the statement is incorrect.
There is at least one mistake.
For example, choice (d) should be False.
d dx cosx + c = sinx so the statement is incorrect.
There is at least one mistake.
For example, choice (e) should be True.
d dx sinx + c = cosx
Correct!
  1. False 2x is the derivative of x2.
  2. True d dx 1 2x4 + c = 4 2x3 = 2x3
  3. False d dx 1 3x3 + 3x2 + x + c = x2 + 6x + 1 so the statement is incorrect.
  4. False d dx cosx + c = sinx so the statement is incorrect.
  5. True d dx sinx + c = cosx
Which of the following correctly gives the anti-derivative F(x) with F(x) = 2sinx + 2x2 + 5x + 1 where F(0) = 2? Exactly one option must be correct)
a)
F(x) = 2cosx + x3 + 5 2x2 + x
b)
F(x) = 2cosx + 1 3x3 + 1 2x2 + x + 4
c)
F(x) = 2cosx + 2 3x3 + 5 2x2 + x
d)
F(x) = 2cosx + 2 3x3 + 5 2x2 + x + 4

Choice (a) is incorrect
Try again, the derivative of this function is F(x) = 2sinx + 3x2 + 5x + 1.
Choice (b) is incorrect
Try again, the derivative of this function is F(x) = 2sinx + x2 + x + 1.
Choice (c) is incorrect
Try again, F(0) = 2 but the derivative of this function is F(x) = 2sinx + 2x2 + 5x + 1.
Choice (d) is correct!
2sinx + 2x2 + 5x + 1 = 2cosx + 2 3x3 + 5 2x2 + x + c.
Since F(0) = 2 + 0 + 0 + 0 + 0 + c = 2 c = 4 so the function
F(x) = 2cosx + 2 3x3 + 5 2x2 + x + 4
A student wishes to find the area between the curve 8 + 2x x2 and the x-axis and produces the following working. Area =42(8 + 2x x2)dx = 8x + x2 1 3x3 42 = 16 + 4 8 3 32 + 16 64 3 = 4 72 3 = 20 square units. Which of the following statements are correct. Exactly one option must be correct)
a)
The limits of integration are wrong.
b)
The anti-derivative has been calculated incorrectly.
c)
The arithmetic is incorrect.
d)
The answer is correctly worked out at every step.
e)
None of the above.

Choice (a) is correct!
8 + 2x x2 = (4 x)(2 + x) so the limits of integration should be -2 and 4. We then get a positive value for the area as expected.
Area =24(8 + 2x x2)dx = 8x + x2 1 3x3 24 = 32 + 16 64 3 16 + 4 8 3 = 60 72 3 = 36 square units
Choice (b) is incorrect
Try again, the anti-derivative is correct.
Choice (c) is incorrect
Try again, the arithmetic is correct.
Choice (d) is incorrect
Try again, the answer is incorrect.
Choice (e) is incorrect
One of the above is the correct answer.
Which of the following integrals computes the area under the curve y = (x + 1)(x 1)(x 3)? Exactly one option must be correct)
a)
11x3 3x2 x + 3dx +13x3 3x2 x + 3dx
b)
11x3 3x2 x + 3dx 13x3 3x2 x + 3dx
c)
11x3 3x2 x + 3dx 13x3 3x2 x + 3dx
d)
13x3 3x2 x + 3dx 11x3 3x2 x + 3dx

Choice (a) is incorrect
Try again, this gives 13x3 3x x + 3dx which is not the required integral.
Choice (b) is incorrect
Try again, the limits of integration are incorrect.
Choice (c) is correct!
The curve looks like this
PIC
The value of the integral between -1 and 1 is positive and between 1 and 3 is negative so first we need to integrate between -1 and 1 and then, we need to integrate between 1 and 3 and change that integral’s sign. Thus we evaluate
11x3 3x2 x + 3dx 13x3 3x2 x + 3dx
Choice (d) is incorrect
Try again, you may need to sketch the curve.